2,185 research outputs found
Certified dimension reduction in nonlinear Bayesian inverse problems
We propose a dimension reduction technique for Bayesian inverse problems with
nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation
noise. The likelihood function is approximated by a ridge function, i.e., a map
which depends non-trivially only on a few linear combinations of the
parameters. We build this ridge approximation by minimizing an upper bound on
the Kullback--Leibler divergence between the posterior distribution and its
approximation. This bound, obtained via logarithmic Sobolev inequalities,
allows one to certify the error of the posterior approximation. Computing the
bound requires computing the second moment matrix of the gradient of the
log-likelihood function. In practice, a sample-based approximation of the upper
bound is then required. We provide an analysis that enables control of the
posterior approximation error due to this sampling. Numerical and theoretical
comparisons with existing methods illustrate the benefits of the proposed
methodology
A max-plus finite element method for solving finite horizon deterministic optimal control problems
We introduce a max-plus analogue of the Petrov-Galerkin finite element
method, to solve finite horizon deterministic optimal control problems. The
method relies on a max-plus variational formulation, and exploits the
properties of projectors on max-plus semimodules. We obtain a nonlinear
discretized semigroup, corresponding to a zero-sum two players game. We give an
error estimate of order , for a
subclass of problems in dimension 1. We compare our method with a max-plus
based discretization method previously introduced by Fleming and McEneaney.Comment: 13 pages, 5 figure
The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis
We introduce a max-plus analogue of the Petrov-Galerkin finite element method
to solve finite horizon deterministic optimal control problems. The method
relies on a max-plus variational formulation. We show that the error in the sup
norm can be bounded from the difference between the value function and its
projections on max-plus and min-plus semimodules, when the max-plus analogue of
the stiffness matrix is exactly known. In general, the stiffness matrix must be
approximated: this requires approximating the operation of the Lax-Oleinik
semigroup on finite elements. We consider two approximations relying on the
Hamiltonian. We derive a convergence result, in arbitrary dimension, showing
that for a class of problems, the error estimate is of order or , depending on the
choice of the approximation, where and are respectively the
time and space discretization steps. We compare our method with another
max-plus based discretization method previously introduced by Fleming and
McEneaney. We give numerical examples in dimension 1 and 2.Comment: 31 pages, 11 figure
Spectral discretization errors in filtered subspace iteration
We consider filtered subspace iteration for approximating a cluster of
eigenvalues (and its associated eigenspace) of a (possibly unbounded)
selfadjoint operator in a Hilbert space. The algorithm is motivated by a
quadrature approximation of an operator-valued contour integral of the
resolvent. Resolvents on infinite dimensional spaces are discretized in
computable finite-dimensional spaces before the algorithm is applied. This
study focuses on how such discretizations result in errors in the eigenspace
approximations computed by the algorithm. The computed eigenspace is then used
to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff
distance between the computed and exact eigenvalue clusters are obtained in
terms of the discretization parameters within an abstract framework. A
realization of the proposed approach for a model second-order elliptic operator
using a standard finite element discretization of the resolvent is described.
Some numerical experiments are conducted to gauge the sharpness of the
theoretical estimates
Parallel eigensolvers in plane-wave Density Functional Theory
We consider the problem of parallelizing electronic structure computations in
plane-wave Density Functional Theory. Because of the limited scalability of
Fourier transforms, parallelism has to be found at the eigensolver level. We
show how a recently proposed algorithm based on Chebyshev polynomials can scale
into the tens of thousands of processors, outperforming block conjugate
gradient algorithms for large computations
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